In Einstein's special relativity theory we live in 4 dimensional spacetime. Though the way we normally "imagine" the world, we tend to believe that we live in a 3 dimensional Newtonian space with a separate absolute time dimension. In this approximation we think of the Earth as round. But is the Earth still round if we leave this approximation and look at it in 4 dimensional spacetime?
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2 Answers
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In four-dimensional spacetime the earth is roughly a cylinder, or a round bar. The length of the bar is the time axis and each cross section of the bar is a 3D ball shape. To get a sense of the scale, how much more narrow it is than the length, perhaps thinking of a very long piece of string or spaghetti would be appropriate. The diameter is a few thousand km, but the length is billions of light years! That is already about $3 \ 10^{18}$ times as long as it is wide, with no indication that we are currently anywhere near the end of the earth in the time direction (despite how it may feel sometimes with all of the wonders of 2020).
If the earth were round in all 4 dimensions then that would mean that it would start as a point rapidly expand to a sphere the size of the earth and then rapidly contract back to a point and disappear. The duration from start to disappear would be about 40 ms since that is roughly the diameter of the earth divided by c.
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$\begingroup$ Does it even make sense to talk about cylinders in 4 dimensions? To me it would seem you are describing a a flat circle in 2+ 1 dimensions. $\endgroup$– JohnnyCommented Sep 23, 2020 at 17:49
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$\begingroup$ We don't have standard English words to describe 4 dimensional shapes. So that is why I said "roughly" and then described in more detail the shape in the second sentence. A cylinder is a 3 dimensional shape whose cross sections are 2D disks, this is a 4 dimensional shape whose cross sections are 3D balls. So a cylinder is the best English word to describe it. $\endgroup$– DaleCommented Sep 23, 2020 at 17:54
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Because of the relativistic phenomenon of length contraction the shape of an object depends on its velocity relatively to the observer.
If you flied on a very fast spaceship past Earth, you would not see it as round. It would look like an ellipsoid, flatter in the direction of motion.
But in its rest frame, the Earth is still round.
As for the four-dimension thing: locally, we still live in a three-dimensional space with a separate and very different dimension of time.
What (special) relativity says is that space and time are not transformed independently when we change our point of view, and that the single invariant quantity that is the spacetime interval replaces the invariant space and time intervals of Newtonian physics.
answered Mar 1, 2018 at 19:54
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$\begingroup$ I wasn't thinking about length contraction. I think of the passage of time as an illusion. It's a way for us to visualise 4 dimensional spacetime as a 3 dimensional space at different times. So I was thinking about what shape the Earth has in 4 dimensions. $\endgroup$ Commented Mar 1, 2018 at 20:44
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$\begingroup$ Also when I am standing on Earth I am not freely falling, I feel a force on my feet. I am accelerated through spacetime. But a person on the other end of the Earth will say the same which seems like a contradiction. But I guess it's only a contradiction when seen from the Earth restframe? So does it make sense to see the Earth as round? $\endgroup$ Commented Mar 1, 2018 at 20:54
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$\begingroup$ @KasperFalkenbergAndersen. I have no idea what the proper definition of a four-dimensional spaciotemporal "shape" would be, if there is any. Not to mention that I have no idea what kind of observer could "see" this. $\endgroup$ Commented Mar 1, 2018 at 21:59